# Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind

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## Contents

## Theorem

Let $z \in \C$ be a complex number whose real part is positive.

Then:

- $z^{\underline r} = \displaystyle \sum_{k \mathop = 0}^m \left[{r \atop r - k}\right] \left({-1}\right)^k z^{r - k} + \mathcal O \left({z^{r - m - 1} }\right)$

where:

- $\displaystyle \left[{r \atop r - k}\right]$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane
- $z^{\underline r}$ denotes $z$ to the $r$ falling
- $\mathcal O \left({z^{r - m - 1} }\right)$ denotes big-$\mathcal O$ notation.

## Proof

## Historical Note

Donald E. Knuth acknowledges the work of Benjamin Franklin Logan in his *The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed.* of $1997$.

## Sources

- 1992: Donald E. Knuth:
*Two Notes on Notation*(*Amer. Math. Monthly***Vol. 99**: pp. 403 – 422) www.jstor.org/stable/2325085 - 1997: Donald E. Knuth:
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